Moment redistribution effects in beams

Transcription

1 Magazine of Concrete Research, 25, 57, No. 1, February, 9 2 Moment redistribution effects in beams R. H. Scott* and R. T. Whittle University of Durham; Arup Research and Development Moment redistribution in beams has traditionally been considered as an ultimate limit state (ULS) phenomenon closely associated with considerations of reinforcement ductility. This paper demonstrates that a significant proportion of this redistribution will almost always occur at the serviceability limit state (SLS) because of the mismatch between the flexural stiffnesses assumed when calculating moments for the ULS and those actually occurring at the SLS due to variations in the reinforcement layout along the member and the influence of cracking. This is demonstrated analytically in the paper and comments are made concerning the current recommendations in BS 811 concerning member stiffness. Tests on 33 two-span beams are then presented, parameters investigated being values of redistribution, beam depth, reinforcement arrangements, concrete strength and the effect of brittle reinforcement. The results confirm that there is significant redistribution at the SLS and that there is scope for increasing the permissible limits for redistribution beyond those currently prescribed in design codes. This is supported by further modelling although it is shown that considerations of crack width may become a limiting factor. Introduction Normal practice is to use a linear elastic analysis for calculating the bending moment and shear force distributions in a reinforced concrete structure. This has the virtue of simplicity as well as permitting results from a series of analyses to be combined using the principle of superposition. It is endorsed by major design codes such as BS 811, 1 EC 2 2 and ACI The assumption of linear elastic behaviour is reasonable at low levels of loading but it becomes increasingly invalid at higher loads due to cracking and the development of plastic deformations. Once an element cracks the behaviour becomes non-linear but it is still reasonable to assume that the tension reinforcement and the concrete in compression both behave elastically up to yield of the reinforcement. Design codes permit elastic analysis to be used at the ultimate limit state (ULS) but acknowledge this nonlinear behaviour by allowing a limited amount of moment redistribution from one part of the structure to another. The permissible moment redistribution is linked to the ductility of the reinforcement at the ULS. 4 * University of Durham, South Road, Durham, DH1 3LE, UK Arup Research and Development, 13 Fitzroy Street, London, W1T 4BQ, UK (MCR 128) Paper received 1 December 23; last revised 2 March 24; accepted 5 May 24 Moment redistribution is useful for practical design as it allows some flexibility in the arrangement of reinforcement. It can be used to transfer moment away from congested areas (e.g. beam column connections) into less congested areas (e.g. mid-spans of beams) or help to allow standard reinforcement layouts where small differences occur in the bending moment distributions for a series of beams, thus avoiding the need to detail each beam separately. In addition, useful economies can be achieved when moment redistribution is applied to different load combinations, resulting in a smaller bending moment envelope which still satisfies equilibrium. Implicit in the current use of moment redistribution is the assumption that sections possess sufficient ductility for the requisite plastic deformations to occur. Design codes achieve this by specifying rules which ensure that the tension steel must have yielded, explicitly in the case of ACI 318 (which specifies a minimum reinforcement strain of 75 microstrain) and implicitly in the case of BS 811 and EC 2 (which link percentage redistribution to neutral axis depth). This leads to the question of whether an upper strain limit should be specified in order to avoid rupturing the reinforcement since, with small neutral axis depths, very high reinforcement strains can be expected. BS imposes a minimum neutral axis depth of. 11d (where d is the effective depth to the tension steel) although this was introduced for the practical # 25 Thomas Telford Ltd

2 Scott and Whittle reason that the top surfaces of beams and slabs are often quite rough and, as a consequence, it was deemed sensible to restrict the lever arm used in design calculations. However, BS 811 s limit has the effect of restricting reinforcement strains to a maximum of 28 microstrain, when making the usual assumption of linear strain distribution across the section. In reality, this value is largely meaningless since gross yield of the reinforcement will have occurred by the time this neutral axis depth has been reached, leading to strains greatly in excess of this nominal value. Consequently, designers have effectively worked on the assumption that the reinforcement will be able to develop whatever level of strain is actually required by a specified neutral axis depth and that failure of a section would always be initiated by crushing of the concrete in compression. The above view was challenged in 1987 by the work of Eligehausen and Langer 5 who investigated the consequences of a finite limit to reinforcement ductility on the failure mode of a section. They demonstrated that reinforcement strain at the ULS could be the controlling parameter in the more lightly reinforced types of members, such as slabs. This prompted considerable discussion concerning the ductility requirements for steel reinforcement. It also raised the question of what upper limit to moment redistribution should be permitted in design codes and whether more stringent limitations should be imposed on the level of reinforcement strains developed at the ULS. This paper presents the results of an investigation which aimed to explore the nature of moment redistribution as load is increased on a structure and thus provide some design guidance on the issues outlined above. The first stage in this process is to examine some of the assumptions made when using the code method of elastic analysis to determine bending moment distributions in a reinforced concrete structure. The initial elastic bending moment diagram thus forms the baseline for the redistribution calculation and any assumptions or approximations made in its determination will directly affect the level of redistribution calculated using the above expression. An elastic analysis is controlled by assumptions concerning the value of flexural stiffness (EI) along the member. Recommendations in design codes vary from, at one extreme, EC 2 2 which is non-specific, to BS which gives three options for calculating the EI value at any particular section (clause ). These are the concrete section (the entire concrete cross section, ignoring the reinforcement), the gross section (the entire concrete cross section with the reinforcement included on the basis of modular ratio) and the transformed section (the compression area of the concrete cross section, ignoring the concrete in tension, combined with the reinforcement on the basis of modular ratio). A further option would be to take the cracked section properties including tension stiffening effects of the concrete. Since the reinforcement is not known until the end of the design process, the most frequently used, by far, in practical design is the concrete section. This has the practical virtue of simplicity (for example, a rectangular continuous beam will have constant flexural stiffness along its entire length) but it also has implications for moment redistribution which, perhaps, may not be fully appreciated. Consider the propped cantilever shown in Fig. 1(a), which has a central point load, and compare the effect of altering the EI value along its length. First, assume that the EI for sagging and hogging moments is the same throughout the member (BS 811 s concrete section approach). Then analyse for a range of situations P L/2 L/2 Flexural stiffness considerations Moment redistribution involves adjustment of the bending moment distribution obtained from a linear elastic analysis with the difference between the redistributed and initial bending moment diagrams being an indicator of the amount of redistribution which has occurred. The percentage of moment redistribution at a section along a beam is calculated as follows:- % redistribution ¼ moment after redistribution moment before redistribution moment before redistribution 3 1 (1) Bending moment: PL (a) EI sagging /EI hogging 1 (%) EI sagging /EI hogging 2 (24%) EI sagging /EI hogging 5 (48%) EI sagging /EI hogging 1 (96%) 5 1 Distance: L (b) Fig 1. Influence of flexural stiffness on moment redistribution. (a) propped cantilever; (b) bending moment distributions 1 Magazine of Concrete Research, 25, 57, No. 1

3 where the EI for sagging moments differs from that for hogging moments. To illustrate the point, ratios of EI sagging /EI hogging of 1, 2, 5 and 1 have been selected and the resulting bending moment distributions are plotted in Fig. 1(b). The bending moment distribution for EI sagging /EI hogging ¼ 1 corresponds, of course, to the situation of constant EI all along the beam and thus acts as a reference line for the other distributions. Increasing the sagging stiffness (applicable in the span) relative to the hogging stiffness (applicable at the support) has the expected effect of shifting moments away from the support and into the span. In the extreme case when a stiffness ratio of 1 is used the behaviour becomes very close to that for a simply supported beam. Thus, relative to the bending moment distribution obtained using the concrete section approach, moment redistribution has occurred in all the other bending moment distributions even though all sections along the member are always behaving in a linearly elastic fashion. The percentages of redistribution obtained at the support, when calculated relative to the constant stiffness distribution, are also shown in Fig. 1(b) in order to illustrate this point. Fig. 1(b) emphasises that it is the relative values of EI that control the bending moment distribution, not the absolute values at any particular section. Any change in the relative EI values will cause moment redistribution to occur. The above has consequences for understanding the behaviour of reinforced concrete beams. If, as is usual, the concrete section is used for the analysis, then the bending moment distribution is that for EI constant all along the beam and this is the situation for which the reinforcement is normally designed at the ULS. It is also the bending moment distribution which will actually be achieved at ultimate. However, in the vast majority of beams, the resulting reinforcement layout will vary along the span. This means that actual EI values based on either the gross or transformed section will change along the span giving relative EI values which are no longer unity. Thus the bending moment distributions actually developed along the beam, even at the serviceability limit state (SLS), will differ from those obtained from analyses that use the concrete section approach. Consequently, moment redistribution will occur even though none was anticipated in the calculations. This redistribution will be very small prior to cracking but after cracking the ratio of EI span :EI support will normally increase with a corresponding increase in the level of redistribution, even though the reinforcement behaves elastically. The magnitude of this redistribution can be quite considerable, as will be demonstrated later. A further complication occurs once the section cracks as tension stiffening then causes the EI to vary with the applied moment. The redistribution which occurs as a result of this mismatch between the actual and assumed EI values will be termed elastic redistribution in this paper. It should be noted that this redistribution is occurring Moment redistribution effects in beams under SLS conditions and that the reinforcement is behaving elastically. After the reinforcement yields further redistribution will occur owing to further changes in the relative values of EI. This additional, post-yield, redistribution will be termed plastic redistribution in this paper. Plastic redistribution is the redistribution with which design engineers have traditionally been familiar and which they implicitly make use of during traditional moment redistribution calculations. Total redistribution at the ULS will actually be the sum of the elastic and plastic components. It will not be all plastic as is commonly assumed at present. Elastic redistribution can nearly always be expected even though none may have been anticipated in the design calculations. This is likely to be inconsistent with the amount of redistribution for which the beam has actually been designed and thus the plastic redistribution that actually occurs will be different from the design value. Consequently, a beam designed for no moment redistribution at the ULS will actually have to undergo plastic redistribution in order to offset the elastic redistribution caused by the supports being more heavily reinforced than the spans and the effects of cracking. An apparently attractive alternative is to use for analysis a distribution of EI which is consistent with the actual stiffness distribution of the reinforcement (BS 811 s transformed section option). To be successful this requires that the reinforcement layout be known prior to undertaking the analysis so that correct EI values can be determined and thus ensure that the calculated bending moment diagram is consistent with the beam s own moments of resistance. This is very difficult to achieve so an alternative is to try an iterative approach. For instance, the beam could first be analysed assuming constant EI (the concrete section approach). The reinforcement is then calculated but this will change the actual distribution of EI. The beam is then re-analysed but the bending moment distribution will now have changed because of the changes in EI and the reinforcement layout must now be adjusted to accommodate the new bending moment distribution. This again changes the EI distribution prompting another analysis and another adjustment of the reinforcement layout. This causes yet another change in the distribution of EI leading to yet another analysis and so on. The procedure can become unstable and is thus unsuitable for practical design purposes. Similar comments can be made concerning use of the gross section. From the above it is suggested that, in spite of the recommendations of clause in BS 811, both the gross section and transformed section approaches are unsuitable for design calculation of bending moment distributions at the ULS. Furthermore, it is suggested that the only practical approach is to use EI values for the concrete section when calculating these moments although it is important to recognise the simplifications and approximations which result from this. The above Magazine of Concrete Research, 25, 57, No. 1 11

4 Scott and Whittle is, of course, not true for the analysis of existing sections where the reinforcement layout is known. A programme of laboratory tests was undertaken to investigate the above points in greater detail. The effects of the following parameters were investigated: (a) depth of section (b) different values of design moment redistribution (c) different arrangements of reinforcement (e.g. large bars versus small bars) (d) different concrete strengths (i.e. use of highstrength concrete for some specimens) (e) the effect of brittle reinforcement [i.e. glass fibre reinforced polymer (GFRP) bars]. Further details are given in the next section. Specimen details and test procedure A total of 33 two-span beams, 5. 2 m long and 3 mm wide, were tested to investigate moment redistribution effects between the centre support and adjacent spans. Depths of 4 mm (series A), 25 mm (series B) and 15 mm (series C, D and E) were used. Each span was 2435 mm long and loaded approximately at its mid-point, as indicated in Fig. 2. The main set of 23 specimens, shown in Table 1, was designed for 3% redistribution from the centre support into the adjacent spans and will be described first. The remaining ten specimens, which investigated other aspects of moment redistribution behaviour, are described later. A consistent coding system was used to reference the specimens. A typical example is B2T12D where B indicates high yield bar reinforcement, 2T12 is the tension reinforcement provided over the centre support and D shows the specimen belongs to series D. An H suffix would indicate that the specimen was made with high-strength concrete e.g. B2T12DH. Other suffices (typically X) are used when specimens were repeated. Each series investigated a different percentage of tension (top) steel at the centre support, values ranging from about. 3% (series E) up to around 1. 7% (series C). However, the several specimens within each series (with the exception of series A which comprised one specimen only) achieved this percentage by using different combinations of bar diameters such as 2T2 or 3T16 or 5T12. This enabled bar diameter effects to be examined since the combinations gave approximately the same total area of tension steel. All other reinforcement (bottom steel at the centre support and top and bottom steel in the spans) was the same for all beams within each series. Some tests were repeated and nominally identical specimens are block shaded in Table 1. Seventeen specimens were cast using normal strength concrete and six with high-strength concrete. The normal strength concrete had a maximum aggregate size of 1 mm, an aggregate : cement ratio of 5. 5 and an overall water : cement ratio of. 6. The average compressive cube strength was MPa (standard deviation 7. 9 MPa) and the average indirect cylinder strength was 3. MPa (standard deviation.2 MPa). The highstrength concrete used 1 mm aggregate and microsilica (in slurry form) in addition to Ordinary Portland Cement (CEM 1) and sand. The overall water : cement ratio was. 25 with adequate workability being achieved by the addition of admixtures. It was commercially batched and delivered by truck. The average compres- Load Load * C A C * * D B D * Fig 2. Diagrammatic layout of test specimens All dimensions in mm. All beams 52 mm long overall 3 mm wide. Depths vary see Tables 1 and 2. For bar diameters see Tables 1 and 2. Bars at locations A and B all 15 mm long. Laps all 3 mm. Bars at A, C, C and B, D, D all in the same horizontal plane. Bars at C, C, D, D cranked on plan at lap positions to achieve this. 3 mm cover to centre of longitudinal bars. Stirrups: Series A and B: T1s at 2 centres, all other specimens R6s at 1 mm centres. * indicates normal exit point for wires in gauged bars see text and Tables 1 and Magazine of Concrete Research, 25, 57, No. 1

5 Moment redistribution effects in beams Table 1. Specimen details: 3% moment redistribution Series Specimen code Overall depth: mm Support reinforcement Span reinforcement Top (A) Bottom (B) Top (C, C9) Bottom (D, D9) A B3T16A** 4 3T16 (.53%) 3T12 3T12 3T2 B B2T2B** 25 2T2 (.93%) 3T12 3T12 3T2 B3T16B** 3T16 (.89%) B3T16BL** 3T16 (.89%) B3T16BLX B5T12B** 3T16 (. 91%) 5T12 (.83%) B2T2BH* 2T2 (.93%) B2T2BHX 2T2 (.95%) B5T12BH* 5T12 (.83%) B5T12BHX 5T12 (. 86%) C B2T2C** 15 2T2 (1.7%) 3T12 3T12 3T2 B3T16C** 3T16 (1.63%) B5T12C** 5T12 (1. 53%) D B2T12D** 15 2T12 (.58%) 3T1 3T1 3T12 B2T12DX B2T12DXX 2T12 (.63%) 2T12 (.63%) B3T1D** 3T1 (.63%) B5T8D** 5T8 (.68%) B2T12DH* 2T12 (.58%) B2T12DHX 2T12 (.63%) E B2T8E** 15 2T8 (. 26%) 2T8 2T8 3T8 B2T8EX 2T8 (.28%) B4T6E** 4T6 (.3%) This table should be read in conjunction with Fig. 2. ** indicates four strain-gauged bars. * indicates two strain gauged bars. Details in italics are for high-strength specimens. Shaded groups of specimens are nominally identical. sive cube strength was 12.4 MPa (standard deviation 1. MPa) and the average indirect cylinder strength was 4. 1 MPa (standard deviation. 5 MPa). Highstrength specimens have the H suffix, as mentioned previously, and are also shown in italics in Table 1. As indicated in Table 1, 15 of the 23 specimens contained strain-gauged reinforcing bars. Specimens with normal strength concrete contained four gauged bars, one each at the locations A, B, C, D shown in Fig. 1 and referred to as rods 1 4, respectively. Specimens with high-strength concrete contained two gauged bars, one each at locations A and D and referred to as rods 1 and 4. Gauged bars had electric resistance strain gauges installed in a central longitudinal duct having a cross-section of 4 mm 3 4 mm in bars of 12 mm dia. and above but which was reduced to 3. 2mm3 3. 2mm in 1 mm dia. bars and to 2.5 mm3 2.5 mm in 6mm and 8 mm dia. bars. Typically, 51 strain gauges were installed in each bar of 1 mm dia. or above but this number was reduced to 3 gauges in the 6 mm and 8 mm dia. bars. Gauge spacing varied from 12. 5mmin zones where rapid changes in strain were expected to 5 mm in less critical areas. Gauges generally had a strain limit of 3% but high elongation gauges having a strain limit of 1 15% were used in areas where high strain levels were anticipated. The wiring normally exited from one end of each bar (see Fig. 2) except for the 8 mm and 1 mm dia. bars where both ends were used. Gauged bars formed part of the main reinforcement in each beam and, wherever possible, the centre bar in a group of three was gauged to maintain symmetry of stiffness across the beam width. Obviously this was not possible in locations where there were only two bars but the effects of the resulting asymmetry were believed to be small. Further details of the gauging technique, which ensures that the bond characteristic between the reinforcement and surrounding concrete is undisturbed, have been published elsewhere. 6 A further ten specimens, shown in Table 2, were tested to investigate other aspects of moment redistribution. Three specimens (prefixed W) used ribbed highyield wires instead of bar reinforcement but were otherwise a repeat of the strain-gauged specimens in series D. Three specimens with bar reinforcement were reinforced for zero redistribution, one for 55% redistribution and two specimens examined the effects of Magazine of Concrete Research, 25, 57, No. 1 13

6 Scott and Whittle Table 2. Details of additional specimens Specimen code % Redistribution Overall depth: mm Support reinforcement Span reinforcement Top (A) Bottom (B) Top (C, C9) Bottom (D, D9) W2T12D** T12 (.58%) 3T1 3T1 3T12 W3T1D** 3 3T1 (. 63%) W5T8D** 3 5T8 (.68%) B3T2B 25 3T2 (1.43%) 3T12 3T12 2T2+1T16 B8T12B 25 8T12 (1. 37%) 3T12 3T12 2T2+1T16 B3T12D 15 3T12 (.94%) 3T1 3T1 2T12+1T8 B2T1D T1 (.44%) 3T1 3T1 2T12+2T1 B4T2B T2 (1.9%) 3T12 3T12 2T16+2T12 B5T2B T2 (2.38%) 3T12 3T12 2T16+1T12 F5F13B No. 13 mm diameter GFRP bars (1.1%) 3T12 3T12 2 No. 25 mm diameter GFRP bars This table should be read in conjunction with Fig. 2. ** indicates four strain-gauged bars. redistributing moments from the spans back to the centre support by 3 and 6%, respectively (shown as negative values in Table 2). The final specimen (F5F13B) used GFRP bars for the main tension reinforcement with five 13 mm dia. bars over the centre support and two 25 mm dia. bars in each span (the only two diameters readily obtainable by the authors at the time). This gave the closest possible match to specimen B5T12B with the available materials. Typical stress strain relationships for all the reinforcement types for strains up to 14 microstrain are shown in Fig. 3. The test procedure was to load the beams incrementally until failure occurred. Applied loads, support reactions and reinforcement strain gauge readings (when appropriate) were recorded at every load stage. Surface strains were measured on one side face of all specimens, at selected load stages, using a Demec gauge and a grillage of Demec points. Moment redistribution behaviour Results from the test programme are summarised in Table 3, for specimens designed for 3% redistribution, and Table 4 for the additional specimens. All quoted values of redistribution are calculated relative to the uniform EI case (i.e. an analysis based on the stiffness of the concrete section). The specimens in Table 3 will be discussed first. Specimens designed for 3% moment redistribution The test results confirmed that a number of discrete stages could be identified in moment redistribution Stress: MPa E Steel 2 GPa E GFRP 5 GPa T8 Bar T1 & T12 Bars T16 Bar T2 Bar T8 & T1 Wires T12 Wire GFRP Strain: microstrain Fig. 3. Reinforcement stress strain relationships 14 Magazine of Concrete Research, 25, 57, No. 1

7 Moment redistribution effects in beams Table 3. Results for specimens designed for 3% moment redistribution Series Specimen code Flexural failure Measured % redistribution Max. rod 1 strain: microstrain Elastic Total A B3T16A Ungauged B B2T2B B3T16B B3T16BL B3T16BLX 2 Ungauged B5T12B B2T2BH Yes B2T2BHX Yes Ungauged B5T12BH B5T12BHX Ungauged C B2T2C B3T16C B5T12C D B2T12D Yes B2T12DX Yes Ungauged B2T12DXX Yes Ungauged B3T1D Yes B5T8D Yes B2T12DH Yes B2T12DHX Yes Ungauged E B2T8E Yes B2T8EX Yes Ungauged B4T6E Yes Details in italics are for high-strength specimens. Shaded groups of specimens are nominally identical. Table 4. Results for additional specimens Specimen code % Designed redistribution Flexural failure Measured % redistribution Max. rod 1 strain: microstrain Elastic Total W2T12D 3 Yes W3T1D 3 Yes W5T8D 3 Yes B3T2B 1 13 Ungauged B8T12B 3 2 Ungauged B3T12D Yes 18 Ungauged B2T1D 55 Yes Ungauged B4T2B 3 5 Ungauged B5T2B Ungauged F5F13B Ungauged behaviour, as illustrated in Fig. 4 for the strain-gauged specimen B2T12DH. This figure shows the relationship between percentage redistribution at the centre support and experimental bending moment at the centre support over the full load history of this specimen. From the start of the test until point A (see Fig. 4) the specimen was uncracked, the small values of redistribution being due to the influence of the different percentages of tension steel along the beam on the values for the uncracked EI. The load increment after Magazine of Concrete Research, 25, 57, No. 1 15

8 Scott and Whittle % Redistribution 45 G 4 35 D 3 E C 25 2 F 15 B 1 5 A Experimental bending moment: knm Fig. 4. Development of moment redistribution in specimen B2T12DH point A caused a crack to form at the centre support and the strains in the top tension bars at this location then increased with succeeding load stages until point D was reached. The measured strain distributions in rod 1 over the centre support (location A in Fig. 2) for these load stages are shown in Fig. 5, with curves A to D in Fig. 5 corresponding to points A to D in Fig. 4. Fig. 5 clearly shows how the crack developed between load stages A and D, causing a significant reduction in the support EI relative to that in the span and hence the step change in redistribution behaviour observable in Fig. 4. The span cracked at point D which reduced the span EI of the beam. Between D and E a stable crack pattern developed, with all reinforcement strains elastic, and this zone corresponded to the SLS. Point E marks the onset of yield (i.e. initial deviance from linear behaviour) in the reinforcement at the centre support and at point F a similar situation occurred in the span, by which time the strain in rod 1 had reached around 5 microstrain. F to G was a region of rapid strain development in both locations leading to plastic hinges being formed. The resulting flexural failure was characterised by an inability to maintain applied loads due to the excessive deflections which occurred as a result of the mechanism action. Strains had reached around 42 microstrain and 18 microstrain at the support and span locations respectively (rods 1 and 4) by the end of the test. The final strain distribution in rod 1 at the centre support is shown in Fig. 6. This indicates Strain: microstrain Distance: mm Fig. 6. Specimen B2T12DH: ultimate strain distribution along rod 1 the very localised nature of the gross yield associated with plastic hinge development. Specimen B2T12DH achieved around 4% redistribution by the end of the test. However, it is important to note that some 25% redistribution was achieved when the reinforcement was still behaving elastically; that is, of the 4% total, 25% was elastic redistribution and only 15% was plastic redistribution. A similar pattern of behaviour was observed with the five specimens of series D designed for 3% redistribution, which are all plotted in Fig. 7. These specimens all achieved the designed 3% redistribution at the ULS (albeit only just with B2T12D but comfortably so with the others) with the elastic contribution ranging from around 15% up to about 35%, this spread of values being caused by variations in the nature of the crack patterns in the individual specimens. These specimens all exhibited flexural failures. Ductility limitations were not an issue at any stage during the test programme. None of the reinforcing bars, which were all of UK origin, failed in spite of the very high strain levels reached in some instances. High levels of redistribution were also achieved even when the reinforcement was still behaving elastically at failure. B2T2B failed in shear with a peak strain in rod 1 of only 225 microstrain yet it achieved 25% redistribution at failure, all of which was elastic. The strain distributions for rod 1 in this specimen, shown in Fig. 8, indicate the flatter spread of strains resulting from the absence of gross yield. As Beeby has pointed out, 4 rotations associated with strain distributions such Strain: microstrain Distance: mm Fig. 5. Specimen B2T12DH: early strain distributions along rod 1 A B C D % Redistribution B2T12D B2T12DX 1 B2T12DXX B2T12DH B2T12DHX Experimental bending moment: knm Fig. 7. Moment redistributions for beams of type B2T12D 16 Magazine of Concrete Research, 25, 57, No. 1

9 Moment redistribution effects in beams Strain: microstrain kn 61 kn 81 kn 11 kn 126 kn 176 kn Distance: mm Fig. 8. Specimen B2T2B: strain distributions for rod 1 as those shown in Fig. 8 can be comparable with those due to the much more localised effects of gross yield, as shown in Fig. 6. Figure 9 shows redistribution plots for a range of B series specimens including B2T2B and the FRP specimen F5F13B (Table 4). The two high-strength specimens B2T2BH and B2T2BHX had flexural failures but the others all failed in shear. Specimens B2T2B, B2T2BH and B2T2BHX (which all had two T2 bars as the main support tension steel) behaved very similarly over the elastic range, achieving 25 33% redistribution. These tests, supported by others in the test programme, indicated that concrete strength was not a significant parameter as far as elastic behaviour was concerned (although specimens made with highstrength concrete achieved a higher failure moment due to their larger moment of resistance). Bar diameter may have had an influence on behaviour in the elastic range, as shown by B5T12B in Fig. 9, due to variations in crack pattern resulting from using several small diameter bars rather than a few bars with larger diameter. However, this was not conclusive and there were no discernible differences in redistribution percentages achieved at the ULS between specimens having similar reinforcement percentages but different bar combinations (e.g. 2T2s, 3T16s or 5T12s). An interesting result is that for the FRP specimen F5F13B. This first cracked in the span, leading to high early levels of redistribution (see Fig. 9) but then, as further cracks developed along the specimen, it closely followed the behaviour of B5T12B. F5F13B had achieved 25% redistribution when it failed even though its main tension reinforcement had stress strain characteristics which were entirely linearly elastic. Specimens of series E, which had a low percentage of tension steel over the centre support (typically. 28%), showed very high levels of redistribution very early in the tests. This is illustrated in Fig. 1 for specimen B2T8E. When the support cracked (point A in Fig. 1), the resulting 9% redistribution meant that the beam effectively behaved as two simply supported spans until cracks in the span (initiated at point B) caused moment to be transferred back to the centre support. Both support and span tension steel were elastic until point C was reached. To summarise, specimens which failed in flexure virtually all achieved the designed 3% total redistribution by the end of the test. Levels of elastic redistribution varied from 1% up to around 3% with the 1 15% range being typical even in those specimens that failed in shear. Additional specimens The results for the additional specimens shown in Table 4 will now be discussed (except for F5F13B already dealt with above). Specimens designed for zero percentage of redistribution (B3T2B, B8T12B, B3T12D) had slightly more reinforcement over the centre support than in the span (see Table 2). On first loading, these specimens cracked at the centre support (the location of the largest bending moment) which caused elastic redistribution into the spans of up to 16%, as illustrated in Fig. 11 for specimen B3T12D. With B3T2B the reinforcement remained elastic until a shear failure occurred and there was no reduction in the 1% elastic redistribution achieved in this specimen. Plastic redistribution was needed to offset the elastic redistribution if zero redistribution at the ULS was to be achieved. This was the case with B3T12D (Fig. 11), which failed in flexure, but not so with B3T2B and B8T12B which both failed in shear. This supports the point made earlier that plastic redistribution must occur if specimens designed 4 1 A % Redistribution B2T2B B2T2BH B2T2BHX B5T13B B5T12B % Redistribution B C Experimental bending moment: knm Experimental bending moment: kn m Fig. 9. Moment redistributions for beams of type B2T2B Fig. 1. Moment redistribution for specimenb2t8e Magazine of Concrete Research, 25, 57, No. 1 17

10 Scott and Whittle % Redistribution B3T12D B3T1D Experimental bending moment: knm Fig. 11. Beams with zero and 55% redistribution for zero redistribution are actually to achieve this situation in practice. Results for the single specimen designed for 55% redistribution are also shown in Fig. 11. This specimen actually achieved 5% redistribution at the ULS plus the quite remarkable value of 35% at the SLS. Specimens B4T2B and B5T2B were designed to redistribute moments from the span to the support, by 3 and 6%, respectively. The results for these two specimens are shown in Fig. 12 with negative values indicating that support moments were being increased. Early cracking over the support meant that moments were initially being redistributed into the adjacent spans. As span cracking developed, moments were transferred back to the supports, as intended, but neither specimen achieved anywhere near its designed level of redistribution at the ULS. Both specimens failed in shear, which probably accounts both for the rapid reduction in redistribution at the end of the tests and for the tendency to transfer moment back into the spans, but neither specimen ever seemed likely to achieve its design value, which is a potential cause for concern. However, the results from these two tests are certainly not conclusive and further work in this area would be helpful. The specimens reinforced with ribbed wire (W2T12D, W3T1D, W5T8D) behaved very similarly to the equivalent bar specimens of series D. Design considerations The test results confirmed that total moment redistribution has both elastic and plastic components with the former making a significant contribution to the whole. The results also suggested that total redistribution was not limited by considerations of (UK) reinforcement ductility. Consequently, there is a case for increasing the current 3% design limit on redistribution prescribed in BS and EC 2 2 and the 2% limit in ACI by recognising the contribution made by elastic redistribution which, at present, is a matter not addressed by the design codes. Elastic redistribution occurs at the SLS due to the mismatch between the assumption, for purposes of analysis, of a constant EI (flexural stiffness) value along the member and the actual EI values which are a consequence of the designed reinforcement layout. To explore this further, a programme of analysis was undertaken to find how much elastic redistribution might be expected (from supports to adjacent spans no attempt was made to model the effects of redistribution from spans to supports) for a range of EI span :EI support stiffness ratios. (The actual value of this ratio for a particular beam will be dependent on the chosen reinforcement layout). The following geometry/loading combinations were analysed to cover situations that approximated to the end span and interior span conditions of a continuous beam. (a) Propped cantilever, central point load (simulating the test beams); (b) propped cantilever, uniformly distributed load over whole span; (c) fixed-ended beam, central point load; (d) fixed-ended beam, uniformly distributed load over whole span. The resulting relationships are shown in Fig. 13. Positive values confirm that the elastic contribution can be quite considerable. Negative values indicate that, at the SLS, redistribution will occur from the span to the 4 % Redistribution B4T2B ( 3%) B5T2B ( 6%) Experimental bending moment: kn m Fig. 12. Beams designed for 3 and 6% redistribution Elastic % redistribution Propped : Central point load Propped : u.d.l Fixed : Central point load Fixed : u.d.l EI span /EI support Fig. 13. Relationship between elastic percentage redistribution and flexural stiffness ratio. u.d.l., Uniformly distributed load 18 Magazine of Concrete Research, 25, 57, No. 1

11 Moment redistribution effects in beams supports even though the beam was designed for redistribution at the ULS to be in the opposite direction a sort of reverse redistribution effect. As discussed earlier, even beams designed for zero redistribution will exhibit elastic redistribution from the spans to the supports at the SLS and thus must undergo plastic redistribution to achieve the designed % at the ULS. This reverse redistribution at the SLS could be considered undesirable since the beam is being forced to behave in different ways at the SLS and the ULS. Reverse redistribution can be avoided, or at least reduced, by the level of designed redistribution which is selected, should this be considered desirable. The lower and upper ends of each curve in Fig. 13 correspond approximately to % (i.e. support(s) fully fixed) and 6% total redistribution respectively. Thus the figure indicates that, for a propped cantilever, designing for 6% total redistribution will give about 35% elastic redistribution for the case of central point load or about 2% elastic redistribution in the case of a uniformly distributed load. Rather more comprehensive relationships between elastic redistribution and total redistribution were also developed as part of the modelling exercise and these are shown in Fig. 14. The curve for the propped cantilever/central point load case is consistent with the test results but, overall, the curves should be treated as approximate and further work is needed were they to be used in design charts. Fig. 14 does, however, give a good and realistic overview of the elastic/total redistribution relationships for the four conditions considered and reinforces the case that elastic redistribution is often a significant proportion of the whole. It also reinforces the points made above concerning reverse redistribution. Commonly, of course, support moments, such as those in a continuous beam, are less than those pertaining to the fixed end case. This will lead to greater span moments, an increase in the EI span :EI support ratio and hence more elastic redistribution. Thus the curves in Fig. 14 are likely to be conservative in this situation. It is important to ensure that, whatever upper limit is adopted, the reinforcement remains elastic under service loads and that crack widths at service loads are kept within acceptable limits. Consequently, a further series of analyses was undertaken to investigate these two issues. The beam geometry and loading arrangement modelled were the same as those used in the test beams. The procedure was to choose a reinforcement layout for the midspan sections of the beam and then, for a given level of design redistribution, calculate the reinforcement at the central support. The moment curvature relationships for the span and support locations were then obtained for concretes having compressive cube strengths of 35 and 1 MPa. The applied loads on the beam were then increased from zero through to the ULS and the bending moment distribution and hence percentage redistribution determined at each load stage. Crack widths were also calculated at the central support and at midspan. Two SLS measures were considered. The first was that the structure must not have reached first yield at service load, since the consequent deformations would be unacceptable. A ratio of service load to yield load of. 9 was assumed to be acceptable. Second, the BS 811 crack width limit of. 3 mm was adopted. Figure 15 shows the relationship between service load/yield load and % redistribution for beam depths and concrete strengths consistent with those used in the test programme. Test results for beams with gauged reinforcement are also included and the agreement is good. Fig. 15 indicates that over 5% redistribution is possible before yield at the SLS occurs, except for high-strength beams 15 mm deep for which a limit of 45% would seem appropriate. Midspan crack widths at the SLS were less than. 3 mm in all cases but, over the supports, they rose sharply to unacceptable limits as the yield load was approached for redistribution levels greater than 4%. Overall, for the geometry and loading of the test beams a limit of 4% redistribution would seem acceptable and this could be increased to 45% if crack widths were not an issue. More work is required to investigate crack width considerations in greater detail before specific design recommendations can be made covering a range of geometries and load combinations. At this stage it is still necessary to consider each situation on its individual merits. The work described in this paper indicates that there Elastic % redistribution Propped : Central point load Propped : u.d.l Fixed : Central point load Fixed: u.d.l Designed % redistribution Service load/yield load: % % 8 6 Experimental 25 mm beam (C35 concrete) 4 data points 15 mm beam (C35 concrete) 25 mm beam (C1 concrete) 2 15 mm beam (C1 concrete) % Design redistribution in beam Fig. 14. Relationships between elastic and designed percentage redistribution. u.d.l., Uniformly distributed load Fig. 15. Ratio of service load to yield load versus percentage design redistribution Magazine of Concrete Research, 25, 57, No. 1 19

12 Scott and Whittle is no case for reducing the maximum permitted 3% limit to moment redistribution currently allowed in BS 811 and EC 2. Furthermore, the limit of 2% given in ACI 318 would appear to be over-conservative. It seems reasonable that the upper limit can be increased beyond 3% by taking into consideration the contribution of elastic redistribution and, with total redistribution being the sum of the elastic and plastic components, it is suggested that the 3% limit (together with the current rules limiting neutral axis depth) now be applied just to the plastic component. The absolute upper limit to total redistribution will depend on beam geometry and loading arrangement but could now be as high as 55%. However, crack width considerations and avoidance of reinforcement yield at the SLS are likely to become limiting criteria. Nevertheless, there will frequently be scope for design engineers safely to exceed the current 3% limit if they so wish. (g) No failure of the reinforcing bars occurred in any of the tests which indicates that there was always sufficient ductility in the UK reinforcement. (h) There was no evidence that the maximum permissible redistribution limits given in BS 811, EC 2 and ACI 318 should be reduced. The limit given in BS 811 relating to the neutral axis depth is only related to plastic redistribution and 3% total redistribution should always be possible. However, there is now a strong case for increasing this value although it is not now possible to specify a unique upper limit since the contribution of elastic redistribution is dependent on support conditions and loading arrangement. It should also be noted that, when current limits are exceeded, crack widths and the possibility of yield at the SLS should be checked. Conclusions (a) An investigation of moment redistribution effects was undertaken comprising laboratory tests on 33 two-span beams supported by programmes of numerical modelling. (b) When carrying out elastic analysis at the ULS the most sensible practical approach is to use the EI values for the concrete section, as defined in BS 811. (c) Total moment redistribution has two components. These are elastic redistribution, which occurs because of the non-uniformity of EI and plastic redistribution which occurs after yield of the tension steel. Elastic redistribution is caused by a mismatch between the uniform flexural stiffness assumed and the stiffness values which actually occur due to variations in the reinforcement layout along the member and the influence of cracking. The contribution of elastic redistribution is significant even in members which fail in shear. (d) A consequence of elastic redistribution is that beams designed for zero redistribution will, in fact, undergo plastic redistribution before the ULS is reached. (e) Reinforcement arrangement (e.g. large bars versus small bars) has little effect on total redistribution. However, the percentage of elastic redistribution may be affected owing to variations in the distributions of cracks. ( f ) Concrete strength can influence total redistribution at the ULS since the moment of resistance of the section is increased. However, at the SLS, the influence seems minimal. Acknowledgements Funding for this project was provided by the Engineering and Physical Sciences Research Council with additional support from Arup Research and Development. The strain-gauged beams were tested by Dr A. R. Azizi and the crack width modelling was undertaken by Mr C. Pope of Arup R&D. In addition, the assistance of the following is also gratefully acknowledged: Tarmac Topmix Ltd, Professor A. W. Beeby FREng, the technical staff of the University of Durham and several generations of project students. References 1. British Standards Institution. BS 811: Structural Use of Concrete Part 1: 1997: Code of Practice for Design and Construction. British Standards Institution, London, British Standards Institution. ENV Eurocode 2: Design of Concrete Structures. Part 1-1: General Rules and Rules for Buildings. British Standards Institution, London, American Concrete Institute. ACI 318: Building Code Requirements for Structural Concrete. American Concrete Institute, Farmington Hills, Michigan, USA, Beeby A. W. Ductility in reinforced concrete: why is it needed and how is it achieved? The Structural Engineer, 1997, 75, No. 18, Eligehausen R. and Langer P. Rotation Capacity of Plastic Hinges and Allowable Degree of Moment Redistribution. Research Report, University of Stuttgart, Scott R. H. Intrinsic mechanisms in reinforced concrete beamcolumn connection behaviour. ACI Structural Journal, 1996, 93, No. 3, Discussion contributions on this paper should reach the editor by 1 August 25 2 Magazine of Concrete Research, 25, 57, No. 1